The Burgers superprocess

We define the Burgers superprocess to be the solution of the stochastic partial differential equation ∂∂tu(t,x)=Δu(t,x)−λu(t,x)∇u(t,x)+γu(t,x)W(dt,dx), where t≥0t≥0, x∈Rx∈R, and WW is space-time white noise. Taking γ=0γ=0 gives the classic Burgers equation, an important, non-linear, partial differential equation. Taking λ=0λ=0 gives the super-Brownian motion, an important, measure valued, stochastic process. The combination gives a new process which can be viewed as a superprocess with singular interactions. We prove the existence of a solution to this equation and its Hölder continuity, and discuss (but cannot prove) uniqueness of the solution.